SELECTED TOPICS ON BOOTSTRAP OR BACK TO STATISTICAL

SELECTED TOPICS ON BOOTSTRAP (OR: BACK TO STATISTICAL THEORY) DANA KANER M. SC. SEMINAR IN STATISTICS, JUNE 2017

APPROXIMATE LIKELIHOODS - INTRODUCTION

THE EMPIRICAL LIKELIHOOD

THE EMPIRICAL LIKELIHOOD - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

THE EMPIRICAL LIKELIHOOD - GOOD TO KNOW

APPROXIMATE PIVOT METHODS

APPROXIMATE PIVOT METHODS - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

BOOTSTRAP PARTIAL LIKELIHOOD

BOOTSTRAP PARTIAL LIKELIHOOD - EXAMPLE likelihood 1. 0 0. 8 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

SUMMARY § Warning: It is important to note that in general, non of the described methods produce a true likelihood: a function that is proportional to the probability of a fixed event in the sample space. 1. 0 0. 8 likelihood 0. 6 0. 4 0. 2 200 400 600 800 1000 1200

PARAMETRIC AND NON-PARAMETRIC INFERENCE “Exact” Nonparametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Parametric Bootstrap Fisher information Parametric delta

INFLUENCE FUNCTIONS

JK , IJ & NON PARAMETRIC BS

JK , IJ & NON PARAMETRIC BS “Exact” Nonparametric Bootstrap Parametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Fisher information Parametric delta

THE PARAMETRIC BOOTSTRAP FOR MLE

THE PARAMETRIC BOOTSTRAP - EXAMPLE Normal model 10 0 50 0 20 40 60 80 100 Exponential model 300 200 100 0 20 40 60 80 100 140

PARAMETRIC MAX. LIKELIHOOD INFERENCE - THE BASICS

PARAMETRIC ML & BS INFERENCE, JACKKNIFE

PARAMETRIC ML, PARAMETRIC BS, IJ “Exact” Nonparametric Bootstrap Parametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Fisher information Parametric delta

THE DELTA METHOD

THE DELTA METHOD FOR THE MEAN – EXAMPLE

THE DELTA METHOD’S CONNECTIONS “Exact” Nonparametric Bootstrap Approximate Jackknife Infinitesimal Jackknife Nonparametric delta Parametric Bootstrap Fisher information Parametric delta

THE EXPONENTIAL FAMILY

PROOF – THE PARAMETRIC CASE

Nonparametric Bootstrap Jackknife Infinitesimal Jackknife Nonparametric delta Parametric BS Fisher information Parametric delta